Consider the expanding product variety model of Grossman & Helpman (1991), given an equilibrium price for the intermediate sector equal to $w/\alpha$, how do the authors get the following result for the brand operating profits?
$\pi = \frac{1-\alpha}{n}$ , where $n$ is the number of varieties
Model settings:
Consumption side of the economy
$U_t = \left(\int_{t}^{\infty} e^{-\rho(\tau-t)} logD(\tau) \; d\tau\right) ~~~ [1]$
with $D(\tau)$ reflecting the household's tastes for diversity in consumption and these tastes will generate demands for differentiated products. Let us assume that at each instant in time the number of varieties available in the economy is in the interval $[0, n(t)]$, with $n(t)$ being the number of available varieties at $t$.
We impose the following specification for D such that we have constant elasticity of substitution between every pair of goods:
$D = \left[\int_{0}^{n} x(j)^{\alpha} dj\right]^{1/\alpha}~~~ [2]$, with $~~~0<\alpha<1$, and this makes goods gross substitutes since EoS>1
Then, an household spending $E$ maximizes instantaneous utility by purchasing $x(j)$ units of a variety $j$, with (by using Dixit-Stiglitz lite, i.e. lover for variety):
$x(j) = \frac{Ep(j)^{-\epsilon}}{\left[\int_{0}^{n} p(j')^{1-\epsilon} dj'\right]} ~~~ [3]$
Now, interpret $D$ as single homogenous consumption good consumed by the household, and $D$ is consumed competitively according to technology given by [2]
The equilibrium price $p_D$ is equal to:
$\lambda= p_D =\left[\int_{0}^{n} p(j)^{1-\epsilon} dj\right]^{\frac{1}{1-\epsilon}} ~~~ [4]$
The demand for input $j$ by a firm that manufactures $D$ units of the final good, by using Shephard's Lemma, is given by:
$x(j) = D p(j)^{-\epsilon} \left[\int_{0}^{n} p(j')^{1-\epsilon} dj'\right]^{-\frac{1}{\alpha}} ~~~ [5]$
Equilibrium condition in the market of the final output: $D=E/p_D ~~~[5.1]$
Then, given $[2]$, $x(j)=x$, and thus $D= n^{\frac{1}{\alpha}}x$, and use $X= nx$ to denote the amount of resources embodied in the final good.
Thus, the final output per unit of final input (TFP) is given by:
$D/X = n^{\frac{1-\alpha}{\alpha}}~~~[6]$, and so the productivity of a given stock of resources rises with the number of available varieties.
Then, plugging [5.1] into [1] and maximizing, we get: $\dot{E}/E = r- \rho ~~~ [7]$
Then, normalize $E$ to 1, i.e. $E=1~~~ [8]$
At equilibrium $r=\rho$
Production Variety
Assume production varieties are produced in monopolistic competition, and each firm supplies a unique variety $j$ and one unit of variety can be produced with one unit of labor only.
Thus, the profit function for the variety $j$ is given by:
$\pi(j) = p(j)x(j) -wx(j)$
Equilibrium price: $p(j) = (1/\alpha)w ~~~[10]$
And $\pi = \frac{1-\alpha}{n}$, with $E=1$. Where this guy comes from?
